Moment inequalities for sums of certain independent symmetric random variables

P. Hitczenko; S. Montgomery-Smith; K. Oleszkiewicz

Studia Mathematica (1997)

  • Volume: 123, Issue: 1, page 15-42
  • ISSN: 0039-3223

Abstract

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This paper gives upper and lower bounds for moments of sums of independent random variables ( X k ) which satisfy the condition P ( | X | k ≥ t ) = e x p ( - N k ( t ) ) , where N k are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which N ( t ) = | t | r for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.

How to cite

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Hitczenko, P., Montgomery-Smith, S., and Oleszkiewicz, K.. "Moment inequalities for sums of certain independent symmetric random variables." Studia Mathematica 123.1 (1997): 15-42. <http://eudml.org/doc/216377>.

@article{Hitczenko1997,
abstract = {This paper gives upper and lower bounds for moments of sums of independent random variables $(X_k)$ which satisfy the condition $P(|X|_k ≥ t) = exp(-N_k(t))$, where $N_k$ are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which $N(t) = |t|^r$ for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.},
author = {Hitczenko, P., Montgomery-Smith, S., Oleszkiewicz, K.},
journal = {Studia Mathematica},
keywords = {moment inequalities; sums of independent symmetric random variables},
language = {eng},
number = {1},
pages = {15-42},
title = {Moment inequalities for sums of certain independent symmetric random variables},
url = {http://eudml.org/doc/216377},
volume = {123},
year = {1997},
}

TY - JOUR
AU - Hitczenko, P.
AU - Montgomery-Smith, S.
AU - Oleszkiewicz, K.
TI - Moment inequalities for sums of certain independent symmetric random variables
JO - Studia Mathematica
PY - 1997
VL - 123
IS - 1
SP - 15
EP - 42
AB - This paper gives upper and lower bounds for moments of sums of independent random variables $(X_k)$ which satisfy the condition $P(|X|_k ≥ t) = exp(-N_k(t))$, where $N_k$ are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which $N(t) = |t|^r$ for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.
LA - eng
KW - moment inequalities; sums of independent symmetric random variables
UR - http://eudml.org/doc/216377
ER -

References

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  1. T. Figiel, P. Hitczenko, W. B. Johnson, G. Schechtman, and J. Zinn (1994), Extremal properties of Rademacher functions with applications to Khintchine and Rosenthal inequalities, Trans. Amer. Math. Soc., to appear. Zbl0867.60006
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  3. E. D. Gluskin and S. Kwapień (1995), Tail and moment estimates for sums of independent random variables with logarithmically concave tails, ibid. 114, 303-309. Zbl0834.60050
  4. U. Haagerup (1982), Best constants in the Khintchine's inequality, ibid. 70, 231-283. Zbl0501.46015
  5. M. G. Hahn and M. J. Klass (1995), Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential bound, preprint. 
  6. P. Hitczenko (1993), Domination inequality for martingale transforms of Rademacher sequence, Israel J. Math. 84, 161-178. Zbl0781.60037
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  15. A. W. Marshall and I. Olkin (1979), Inequalities: Theory of Majorization and Its Applications, Academic Press, New York. Zbl0437.26007
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  17. S. J. Montgomery-Smith (1992), Comparison of Orlicz-Lorentz spaces, Studia Math. 103, 161-189. Zbl0814.46023
  18. I. Pinelis (1994), Extremal probabilistic problems and Hotelling’s T 2 test under a symmetry condition, Ann. Statist. 22, 357-368. Zbl0812.62065
  19. I. Pinelis (1994), Optimum bounds for the distributions of martingales in Banach spaces, Ann. Probab. 22, 1679-1706. Zbl0836.60015
  20. H. P. Rosenthal (1970), On the subspaces of L p (p>2) spanned by sequences of independent random variables, Israel J. Math. 8, 273-303. Zbl0213.19303
  21. G. Schechtman and J. Zinn (1990), On the volume of the intersection of two L p n balls, Proc. Amer. Math. Soc. 110, 217-224. Zbl0704.60017
  22. M. Talagrand (1989), Isoperimetry and integrability of the sum of independent Banach-space valued random variables, Ann. Probab. 17, 1546-1570. Zbl0692.60016
  23. S. A. Utev (1985), Extermal problems in moment inequalities, in: Limit Theorems in Probability Theory, Trudy Inst. Mat., Novosibirsk, 1985, 56-75 (in Russian). 

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